Make slaw: Pulse cabbage, in batches if necessary, in a food processor until finely chopped, then transfer to a bowl and add 3/4 teaspoon salt and liquid from pickled beets. Finely chop beets (with a knife) and stir into cabbage mixture along with vinegar. Let stand, stirring occasionally, 30 minutes to allow flavors to develop.
Make pancakes while slaw stands: Remove skin and fat from duck leg and finely chop them (reserve meat), then cook in a 10-inch heavy skillet over moderately low heat, stirring occasionally, until fat is rendered and skin is golden and crisp, about 10 minutes. Transfer duck skin with a slotted spoon to paper towels to drain, reserving fat in skillet. Meanwhile, remove duck meat from bone, discarding bone, and cut into 1/4-inch dice. Add onion to skillet and cook over moderate heat, stirring occasionally, until browned, 7 to 9 minutes. Combine onion, duck meat, and reserved skin in a large bowl.
Peel potatoes and shred using large holes of a box grater. Squeeze potatoes by handfuls to eliminate excess moisture, then add to duck mixture along with egg, pepper, and remaining 3/4 teaspoon salt, stirring until combined.
Spread 1/2 cup bread crumbs on a sheet of wax paper. Using a scant 1/4 cup potato mixture for each pancake, make 12 mounds on crumbs. Coat mounds with remaining 1/2 cup crumbs and flatten into 3-inch patties.
Heat oil in a 12-inch heavy skillet over moderate heat until hot but not smoking. Using a slotted spatula, gently shake off excess crumbs from each potato pancake, then fry in 2 batches, turning over once, until golden, about 6 minutes per batch. Transfer to paper towels to drain. Serve immediately, with slaw.
Cooks' note: Slaw can be made 1 day ahead and chilled, covered.